38 III METRIC SPACES

(3.8.5) The closure of a set A is the complement of the exterior of A.

The closure of an open ball E(a; r) is contained in the closed ball Bf(a; r),
but may be different from it. If a subset A of the real line is majorized
(resp minorized), sup A (resp inf A) is a cluster point of A, as follows from
(2.3.4).

Due to (3.8.5), the four following properties of cluster points and closure
are read off from those proved in Section 3.7 for interior points and interior,
by using the formulas of boolean algebra:

(3.8.6) For any set A, A is the smallest closed set containing A.

In particular, closed sets are characterized by the relation A = A.

(3.8.7) 7/AcB,AcB.

(3.8.8) For any pair of sets A? B, A u B = A u B.

(3.8.9) In order that a point x be a cluster point of A, a necessary and suf-
ficient condition is that d(x, A) = 0.

(3.8.10) The closure of a set A is the intersection of the open neighborhoods
(A)o/A.

This is only a restatement of (3.8.9).

(3.8.11) In a metric space E, any closed set is the intersection of a decreasing
sequence of open sets; any open set is the union of an increasing sequence of
closed sets.

The first statement is proved by considering the open sets V1/n(A) and the
second follows from the first by considering complements.

(3.8.12) If a cluster point x of A does not belong to A, any neighborhood V
ofx is such that V n A is infinite. discrete space every set is closed.